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• W2077872018 abstract "Let W be an A *-algebra such that any maximal abelian *-subalgebra is regular and such that any quasinilpotent element x in VI satisfies xN=0, with N< oo. Then any irreducible Hilbert space *-representation is at most N-dimensional. If 9X is a C*-algebra, W possesses transcendental quasinilpotent elements if there exists a v E sf with dim = oo. This note is a continuation of [1]. We want to establish an intimate connection between the degree of noncommutativity of a Banach algebra W and the existence of nilpotent elements in W. The proofs are straightforward extensions of the matrix situation and of methods used in [1]. Throughout all Banach algebras are complex Banach algebras with norm I 1, continuous involution * and an auxiliary norm [3] 1 lo which satisfies Ixx*10=IX12. Such Banach algebras are called A*-algebras [3]. In A*-algebras maximal abelian *-subalgebras are necessarily semisimple. We shall further assume that all Banach algebras have a unit, because the adjunction of a unit does not affect the proofs nor the results. A representation 7r always denotes a *-representation on a Hilbert space H,. We say that x E W is n-nilpotent, with n= oo, 1, 2, , if x is quasinilpotent and xr=O but xn-1$O. We begin with a simple lemma, which is probably known. LEMMA. Let 9[ be an A*-algebra such that any x=x* E W has a finite spectrum with at nmost n points; then dim %:t?n2. PROOF. Let 28 be a maximal abelian (semisimple)*-subalgebra of WI. Then the character space Q3 of 23 has at most n elements. Hence any x c 03 has the form x= ji,? A,ei, where the ei are the minimal idempotents of 23. The ei are selfadjoint and it is easy to see that dim ei. ej< 1. THEOREM 1. (i) Let W be an A *-algebra such that any maximal abelian *-subalgebra is regular and such that any quasinilpotent element x in W satisfies xN=0, then an)' irreducible representation r of 9f is at most N dimensional. Received by the editors February 10, 1972. AMS (MOS) subject class fications (1969). Primary 4660, 4665." @default.
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• W2077872018 date "1973-01-01" @default.
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• W2077872018 title "Nilpotent elements in Banach algebras" @default.
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• W2077872018 doi "https://doi.org/10.1090/s0002-9939-1973-0315457-8" @default.
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